Now 'amounts' of Triangles , , together = . [II. Prop. 8. But these make up angles of Figure ; angles of Figure together = ; but angles , together = ; [I. Prop. 1. angles , , together = ; , , are equally inclined to . [I. Prop. 1. Therefore a Pair of Lines &c. Q.E.D. Axioms (continued). 2. If two homogeneous magnitudes be both of them finite: the lesser may be so multiplied, by a finite number, as to exceed the greater. PROP. X. Theorem. [Pg 28] If a Pair of Lines make, with a certain transversal, two interior angles, on the same side of it, which are together less than two right angles, the defect being a finite angle: these Lines are intersectional on that side of the transversal. Let , make with the interior angles , together less than . Make angle equal to angle ; produce to ; from draw at right angles to , and at right angles to . Hence is also at right angles to ; [II. Prop. 9. i. e. , have a common perpendicular; each is equidistant from the other; [II. Prop. 7, Cor. 1. also = the common distance between them. [Pg 29] Now angle is the defect, from , of the sum of the 2 interior angles , ; hence, by hypothesis, it is finite; it may be so multiplied, by a finite number, as to exceed angle . [II. Ax. 2. Call this finite number '.' In , produced if necessary, take -times ; from draw at right angles to ; turn Triangle about into position ; then, about , into position , and so on, till there are such Triangles altogether; and let its final position be . Then angle is -times angle ; i. e. it is greater than angle . Let cut broken-Line at ; and join . Then is greater than , which is greater than , which is greater than ; [Euc. I. 20, 17, 19. is greater than ; but is the same fraction of which is of ; is greater than ; i. e. the distance of , from , is greater than the common distance between and . , are intersectional towards , . Therefore, if a Pair of Lines &c. Q.E.D. [Pg 31] APPENDICES TO PART I. [Pg 32] APPENDIX I. Containing an alternative Axiom, which may be substituted for Axiom 1 at p. 14. Definition. The Segment, cut off, from any Sector of a Circle, by its Chord, may be called its 'outer Segment'; and the Triangle, contained by its Chord and its two Radii, may be called its 'central Triangle.' And, if its Arc be bisected and the point of bisection joined to the ends of its Chord, the isosceles Triangle, so formed, may be called its 'inscribed isosceles Triangle.' PROPOSITIONS. PROP. A. Theorem. [Pg 33] If, in any Sector of a Circle, its Chord be not-less than its Radius: then, in a Sector whose vertical angle is twice as great, its outer Segment is greater than its central Triangle. Let be a Sector whose Chord is not-less than its Radius . Make angle equal to angle ; and join , . Then vertical angle of Sector is twice as great as that of Sector . It shall be proved that its outer Segment is greater than its central Triangle .